# The Empirical Mean is Minimax Optimal for Local Glivenko-Cantelli

arXiv: https://arxiv.org/abs/2410.02835 · HTML: https://mlsanity.com/paper/2410.02835/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 18 references that are in the map.
- First posted on arXiv 2 Oct 2024.

## Builds on (most cited first, 7 of 7)

- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [Outlier Robust Mean Estimation with Subgaussian Rates via Stability](https://mlsanity.com/paper/2007.15618/) (2020)
- [Sub-Gaussian estimators of the mean of a random vector](https://mlsanity.com/paper/1702.00482/) (2017)
- [Mean Estimation with Sub-Gaussian Rates in Polynomial Time](https://mlsanity.com/paper/1809.07425/) (2018)
- [Fast Mean Estimation with Sub-Gaussian Rates](https://mlsanity.com/paper/1902.01998/) (2019)
- [Optimal Mean Estimation without a Variance](https://mlsanity.com/paper/2011.12433/) (2020)
- [Exact Lower Bounds for the Agnostic Probably-Approximately-Correct (PAC) Machine Learning Model](https://mlsanity.com/paper/1606.08920/) (2016)

## Built on by (most cited first, 2 of 2)

- [Uniform Convergence Beyond Glivenko-Cantelli](https://mlsanity.com/paper/2510.21506/) (2025)
- [Improved Distribution Estimation in ℓ_∞](https://mlsanity.com/paper/2605.30509/) (2026)

## Abstract (arXiv)

We revisit the recently introduced Local Glivenko-Cantelli setting, which studies distribution-dependent uniform convergence rates of the Empirical Mean Estimator (EME). In this work, we investigate generalizations of this setting where arbitrary estimators are allowed rather than just the EME. Can a strictly larger class of measures be learned? Can better risk decay rates be obtained? We provide exhaustive answers to these questions, which are both negative, provided the learner is barred from exploiting some infinite-dimensional pathologies. On the other hand, allowing such exploits does lead to a strictly larger class of learnable measures.